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Mathematics /AP Calculus AB: 6.3.3 Logarithmic Differentiation

AP Calculus AB: 6.3.3 Logarithmic Differentiation

Mathematics14 CardsCreated 7 months ago

This content explains the method of logarithmic differentiation, which is used when differentiating functions with variable exponents. By taking the natural logarithm of both sides, variable exponents are transformed into products, making it easier to apply implicit differentiation and find derivatives of complex functions.

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Logarithmic Differentiation

Understand that the Power Rule of Differentiation cannot be applied when the exponent is a variable.
Take the natural log of both sides of an equation to simplify the problem by transforming a variable exponent into a product (of the variable with the natural log function).
Apply logarithmic differentiation to complicated functions to make it easier to find derivatives by transforming exponents to products.

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Key Terms

Term

Definition

Logarithmic Differentiation

Understand that the Power Rule of Differentiation cannot be applied when the exponent is a variable.
Take the natural log of...

note

The Power Rule of Differentiation, d/dx(x n ) = nx^n − 1 , may only be applied when the exponent, n, is a fixed number. If there is a v...

Differentiate the given function.

f(x)=x^x+1

f′(x)=x^x(xlnx+x+1)

Differentiate the given function.

f(x)=(2x+3)^5x−7

f′(x)=(2x+3)^5x−8 ((10x+15)ln(2x+3)+10x−14)

Differentiate the given function.

f(x)=(sinx)x^3x

f′(x)=(sinx)x^3x (cotx+3lnx+3)`

Differentiate the given function.

f(x)=x^tanx

f′(x)=x^(tanx)−1(x(sec2x)lnx+tanx)

Differentiate the given function.

f (x) = x^2x

f ′( x) = 2x^2x (ln x + 1)

Differentiate the given function.

f (x) = (cos x)^x

f ′( x) = (cos x)^x (ln (cos x) − x tan x)

Differentiate the given function.

f(x)=(cosx)^sinx

f′(x)=(cosx)^sinx ((cosx)ln(cosx)−sinxtanx)

Differentiate the given function.

f(x)=(sinx)^x

f′(x)=(sinx)^x (ln(sinx)+xcotx)

Differentiate the given function.

f(x)=(√7x−1)x^2

f′(x)=(√7x−1)^x^2 ((14x^2−2x)ln(7x−1)+7x^2/14x−2)

Differentiate the given function.

f(x)=x(tanx)^x

f′(x)=(tanx)^x(1+xln(tanx)+x^2cscxsecx)

Differentiate the given function.

f(x)=x^√2x+1

f′(x)=x^√2x+1 (xlnx+2x+1/x√2x+1)

Differentiate the given function.

f(x)=(lnx)^lnx

f′(x)=(lnx)^lnx/x (ln(lnx)+1)

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AP Calculus AB: 6.3.3 Logarithmic Differentiation

Term	Definition
Logarithmic Differentiation	Understand that the Power Rule of Differentiation cannot be applied when the exponent is a variable. Take the natural log of both sides of an equation to simplify the problem by transforming a variable exponent into a product (of the variable with the natural log function). Apply logarithmic differentiation to complicated functions to make it easier to find derivatives by transforming exponents to products.
note	The Power Rule of Differentiation, d/dx(x n ) = nx^n − 1 , may only be applied when the exponent, n, is a fixed number. If there is a variable in the exponent, as with the function y = x^x , the best approach is to turn the exponent into a product by taking the natural log of both sides of the equation. To find the derivative, differentiate the resulting equation implicitly with respect to x. The expression (1/y)·y ́ will always be on the left side of the equation. To solve for y ́, multiply both sides of the equation by y. Then on the right side of the equation, replace y with its equivalent expression in terms of the x that was given at the very beginning of the problem. Apply logarithmic differentiation whenever the function whose derivative is sought has a variable appearing in an exponent. In this example the steps are laid out carefully. First, take the natural log of both sides and pull out cos x in the exponent so that it becomes a term in the product. Second, implicitly differentiate both sides, remembering to apply the Product Rule of Differentiation and the Chain Rule as necessary. Third, multiply both sides by y and replace y with (sin x) cos x .
Differentiate the given function. f(x)=x^x+1	f′(x)=x^x(xlnx+x+1)
Differentiate the given function. f(x)=(2x+3)^5x−7	f′(x)=(2x+3)^5x−8 ((10x+15)ln(2x+3)+10x−14)
Differentiate the given function. f(x)=(sinx)x^3x	f′(x)=(sinx)x^3x (cotx+3lnx+3)`
Differentiate the given function. f(x)=x^tanx	f′(x)=x^(tanx)−1(x(sec2x)lnx+tanx)
Differentiate the given function. f (x) = x^2x	f ′( x) = 2x^2x (ln x + 1)
Differentiate the given function. f (x) = (cos x)^x	f ′( x) = (cos x)^x (ln (cos x) − x tan x)
Differentiate the given function. f(x)=(cosx)^sinx	f′(x)=(cosx)^sinx ((cosx)ln(cosx)−sinxtanx)
Differentiate the given function. f(x)=(sinx)^x	f′(x)=(sinx)^x (ln(sinx)+xcotx)
Differentiate the given function. f(x)=(√7x−1)x^2	f′(x)=(√7x−1)^x^2 ((14x^2−2x)ln(7x−1)+7x^2/14x−2)
Differentiate the given function. f(x)=x(tanx)^x	f′(x)=(tanx)^x(1+xln(tanx)+x^2cscxsecx)
Differentiate the given function. f(x)=x^√2x+1	f′(x)=x^√2x+1 (xlnx+2x+1/x√2x+1)
Differentiate the given function. f(x)=(lnx)^lnx	f′(x)=(lnx)^lnx/x (ln(lnx)+1)